Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian and let $r>0$.

Quick scholar googling gave this reference: jstor.org/stable/2244253, which cites a similar result for general two-dimensional Riemann manifold (though the number 2 is missing from the rhs there).
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zhorasterMar 22 '11 at 15:28

Thanks zhoraster. I'm familiar with these two papers but they focus on the case where $t$ is fixed and $r\to 0$. They essentially proved that in this scenario: $$ \mathbb{E}(\mathrm{vol}(W_{r}(t)))\sim \frac{\pi t}{\log(1/r)}+\frac{\pi t}{2\log(1/r)^2}(1+k-\log(2t)). $$ However, I'm interested in the case where $r$ is fixed and $t\to\infty$ as in the Euclidean case.
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ghtMar 22 '11 at 18:32

BTW, by just comparing with the Euclidean case you see that the behavior is quiet different in these two cases.
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ghtMar 22 '11 at 18:34

1 Answer
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I just found out that the case $r$ fixed and $t\to\infty$ for simply connected symmetric manifolds of non-positive sectional curvature and dimension $d\geq 3$, and strictly negative curvature for dimension $d=2$, was solved by Chavel and Feldman in "The Wiener Sausages and a Theorem of Spitzer in Riemannian Manifolds", Probability and Harmonic Analysis, New York, pp. 45-60, 1986.